Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C2×#10576928 (order 1024) · Inn(G) = D8 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×SD32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D8 |
| commutator, Frattini, cosocle | C8 | C23×C3 |
| prime radical | C2×C3×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(19,20,21);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18);(4,10)(5,17)(6,8)(7,15)(9,13)(12,18)(14,16)66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 939);